This project is to develop new mathematical methods to better model the loop regions of proteins. Loop regions lack secondary structure and predicting their 3-dimensional conformation from amino acid sequences is one of the main challenges in the study of protein structure and function. Loops are often the sites of the biological mechanisms of action of a protein. Learning these biological mechanisms requires mathematical and computational methods that can sample these conformations efficiently. Due to their inherent flexibility, loop regions may assume a vast variety of shapes and discovering the biologically relevant conformations of low free energy by purely random search can be prohibitive. The discovery and efficient incorporation of appropriate constraints can dramatically reduce the conformational search problem and make it tractable to computation. The Coutsias and Dill groups have published collaboratively on these problems for ten years and contributed some of the current state-of-the-art methods to various software. Here it is proposed: (1) to generalize current state of the art methods for imposing loop closure constraints to treat arbitrary steric and other physical or geometrical constraints in a unified formalism;(2) to develop the mathematics more deeply, relating the numerical analysis of constrained loop closure algorithms to the underlying algebraic and geometric properties of multivariate polynomial systems;(3) to combine our static constraint methods with Gaussian Net dynamics methods to treat dynamics efficiently too;(4) to further increase the efficiencies and coverings through the development of novel concerted move sets combined with a deeper understanding of the topological and geometrical properties of constrained conformation spaces, and (5) to apply them to several biologically important loop modeling problems. If successful, the methods developed in this project will be useful for better understanding biological mechanisms of action and for computational drug discovery, where ligand binding to a protein often depends on its interactions with loops. PUBLIC HEALTH RELEVANCE: Reliable computer determination of the structures of loops in proteins has enormous practical applications: It enables not only prediction of loop conformations controlling biological processes - such as antigen recognition, signal transduction, and enzyme active site gating - but also reengineering of loops at critical locations in proteins for new functions.